Geometric genus of a plane curve by delta invariants

Given: The Axiom of Choice and an algebraically closed field $k$, an irreducible homogeneous form $F$ of degree $d\ge 1$, the integral plane curve $X=V_{+}(F)$ with isolated singularities, its normalization $\nu :X^{\mathrm{n}\mathrm{u}}\to X$, and the delta invariants ${\delta }_x(X)$ of its closed points.

Under Choice, for the integral plane curve $X=V_{+}(F)$ of degree $d$ one has $H^0(X,{\mathcal{O}}_X)=k$ and $p_a(X)=1-\chi ({\mathcal{O}}_X)=\frac{(d-1)(d-2)}{2}$. (Arithmetic genus of a plane curve)

Under Choice, for an integral proper finite-type curve over the algebraically closed field $k$ with normalization $\nu :X^{\mathrm{n}\mathrm{u}}\to X$, one has $p_a(X)=g(X^{\mathrm{n}\mathrm{u}})+{\sum }_{x\in X}{\delta }_x(X)$, the sum finite and supported on the singular points; the delta invariant is ${\delta }_x(X)={\dim }_k(({\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}}{)}_x\mathrm{/}{\mathcal{O}}_{X,x})$, and ${\delta }_x(X)=0$ exactly at the regular points. (Arithmetic genus, geometric genus and delta invariants, Delta invariant of a curve singularity)

Under Choice, for the integral plane curve $X=V_{+}(F)$ over the algebraically closed field $k$ the normalization $\nu :X^{\mathrm{n}\mathrm{u}}\to X$ exists, is finite and birational, and the geometric genus is defined as $g(X)=g(X^{\mathrm{n}\mathrm{u}})$. (Normalization of an integral finite-type curve by gluing affine integral closures, Geometric genus of a singular curve)

Under Choice, if $X$ is an integral finite-type $k$-scheme whose underlying space has chain dimension one, then a proper closed subset $Z⊊X$ is a finite set of closed points, and every point other than the generic point is closed. (Proper closed subsets of a curve are finite)

The Axiom of Choice is assumed for the cited interfaces. (The Axiom of Choice)